Team Parsifal

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Section: Scientific Foundations

Keywords : deep inference, category theory.

Deep Inference and Categorical Axiomatizations

Deep inference [33] , [34] is a novel methodology for presenting deductive systems. Unlike traditional formalisms like the sequent calculus, it allows rewriting of formulas deep inside arbitrary contexts. The new freedom for designing inference rules creates a richer proof theory. For example, for systems using deep inference, we have a greater variety of normal forms for proofs than in sequent calculus or natural deduction systems. Another advantage of deep inference systems is the close relationship to categorical proof theory. Due to the deep inference design one can directly read off the morphism from the derivations. There is no need for a counterintuitive translation.

One reason for using categories in proof theory is to give a precise algebraic meaning to the identity of proofs: two proofs are the same if and only if they give rise to the same morphism in the category. Finding the right axioms for the identity of proofs for classical propositional logic has for long been thought to be impossible, due to “Joyal's Paradox”. For the same reasons, it was believed for a long time that it it not possible to have proof nets for classical logic. Nonetheless, Lutz Straßburger and François Lamarche provided proof nets for classical logic in [4] , and analyzed the category theory behind them in [36] . In [10] and [54] , one can find a deeper analysis of the category theoretical axioms for proof identification in classical logic. Particular focus is on the so-called medial rule which plays a central role in the deep inference deductive system for classical logic.

The following research problems are investigated by members of the Parsifal team:


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